On interconnections of discontinuous dynamical systems: an input-to-state stability approach

In this paper we will extend the input-to-state stability (ISS) framework to continuous-time discontinuous dynamical systems (DDS) adopting non-smooth ISS Lyapunov functions. The main motivation for investigating non-smooth ISS Lyapunov functions is the success of "multiple Lyapunov functions" in the stability analysis of hybrid systems. This paper proposes an extension of the well-known Filippov solution concept, that is appropriate for \´open\´ systems so as to allow interconnections of DDS. It is proven that the existence of a non-smooth ISS Lyapunov function for a DDS implies ISS. In addition, a (small gain) ISS interconnection theorem is derived for two DDS that both admit a non-smooth ISS Lyapunov function. This result is constructive in the sense that an explicit ISS Lyapunov function for the interconnected system is given. It is shown how these results can be applied to construct piecewise quadratic ISS Lyapunov functions for piecewise linear systems (including sliding motions) via linear matrix inequalities.